Finite Group Databasebeta

← #30637240

Subgroups — #30637240

Order 16256.

228 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550428C211yesmaximal, characteristic
2#22549930C211yesmaximal, characteristic
3#22549721C211yesmaximal, characteristic
4#22549614C211yesmaximal, characteristic
5#22549613C211yesmaximal, characteristic
6#22549612C211yesmaximal, characteristic
7#328×C127C211yesmaximal, characteristic
8#16861208C411yescharacteristic, complemented
9#16861207C411yescharacteristic, complemented
10#16861199—21no—
11#16861188C2211yescharacteristic
12#16861186—21no—
13Q8×Dic127C411yescharacteristic, complemented
14#16861174—21no—
15#16861174C411yescharacteristic, complemented
16#16861165—21no—
17C2×C4×Dic127—21no—
18C2×C4×Dic127C2211yescharacteristic
19Dic127⋊C8—21no—
20C22⋊(C127⋊C8)C2211yescharacteristic
21#16861119—21no—
22C127⋊#95—21no—
23C127⋊#95C2211yescharacteristic
24C22⋊Q8×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C2×Dic254—21no—
28C2×Dic254C2×C411yescharacteristic
29C22×Dic127—21no—
30C22×Dic127D411yescharacteristic
31C22×Dic127D411yescharacteristic
32C2×C127⋊C8—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127C2×C411yescharacteristic
35Dic127⋊C4—21no—
36Dic127⋊C4—21no—
37Dic127⋊C4—21no—
38Dic127⋊C4—21no—
39Dic127⋊C4—21no—
40C22⋊Dic127—21no—
41C4×Dic127—24no—
42C4×Dic127—21no—
43C4×Dic127—21no—
44C4×Dic127C2×C411yescharacteristic
45C4×Dic127C2×C411yescharacteristic
46C4×Dic127D411yescharacteristic
47C4×Dic127D411yescharacteristic
48C2×Q8×C127C2×C411yescharacteristic
49C22×C4×C127—21no—
50C22×C4×C127C2311yescharacteristic
51C2×C8×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127C2×C411yescharacteristic
54C22⋊C4×C127—21no—
55Dic254—42no—
56Dic254—41no—
57C2×Dic127—21no—
58C2×Dic127—21no—
59C2×Dic127—22no—
60C2×Dic127—22no—
61C2×Dic127—21no—
62C2×Dic127—21no—
63C2×Dic127—41no—
64C2×Dic127—41no—
65C2×Dic127C22⋊C411yescharacteristic
66C2×Dic127C22⋊C411yescharacteristic
67C2×Dic127C22⋊C411yescharacteristic
68C2×Dic127C22⋊C411yescharacteristic
69C127⋊C8—41no—
70C23×C127C2×D411yescharacteristic
71Q8×C127—41no—
72C2×C4×C127—22no—
73C2×C4×C127—21no—
74C2×C4×C127—21no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127C22×C411yescharacteristic
79C2×C4×C127C2×D411yescharacteristic
80C8×C127—41no—
81C22×C127—21no—
82C22×C127—21no—
83C22×C127C2×C22⋊C411yescharacteristic
84C22×C127#13711yescharacteristic
85C22×C127#13711yescharacteristic
86C4×C127—21no—
87C4×C127—21no—
88C4×C127—42no—
89C4×C127—41no—
90C4×C127—41no—
91Dic127—21no—
92Dic127—21no—
93Dic127—21no—
94Dic127—42no—
95Dic127—41no—
96Dic127—41no—
97Dic127#10412yes—
98C2×C127—21no—
99C2×C127—21no—
100C2×C127#41011yescharacteristic
101C2×C127#41911yescharacteristic
102C2×C127C2×#10411yescharacteristic
103#1506—1271nomaximal, Sylow
104C127#150611yescharacteristic, complemented, Sylow
105#533—1271no—
106#432—1271no—
107#376—1271no—
108#328—1271no—
109#328—1271no—
110#328—1271no—
111#328D12711yescharacteristic
112C4⋊Q8—1271no—
113C4⋊Q8—1271no—
114#125—2541no—
115C22⋊Q8—1271no—
116C22⋊Q8—2541no—
117C22⋊Q8C2×D12711yescharacteristic
118C4×Q8—1271no—
119C4×Q8—1271no—
120C4×Q8—2541no—
121#117—2541no—
122C2×C42—1271no—
123C2×C42—2541no—
124C4⋊C8—2541no—
125C22⋊C8—1271no—
126C22⋊C8C2×D12711yescharacteristic
127#97—2541no—
128#95—1271no—
129#95—2541no—
130#95C2×D12711yescharacteristic
131C2×Q8—1271no—
132C2×Q8—2541no—
133C2×Q8C4×D12711yescharacteristic
134C22×C4—21no—
135C22×C4—1271no—
136C22×C4—1271no—
137C22×C4—2541no—
138C22×C4C22×D12711yescharacteristic
139C2×C8—21no—
140C2×C8—2541no—
141C4⋊C4—21no—
142C4⋊C4—1271no—
143C4⋊C4—2541no—
144C4⋊C4—2541no—
145C4⋊C4—2541no—
146C4⋊C4—2541no—
147C4⋊C4—2541no—
148C4⋊C4—2541no—
149C4⋊C4C4×D12711yescharacteristic
150C22⋊C4—21no—
151C22⋊C4—2541no—
152C42—1271no—
153C42—1271no—
154C42—1271no—
155C42—1271no—
156C42—2541no—
157C42—2541no—
158C42—2544no—
159C23D4×D12711yescharacteristic
160Q8—41no—
161Q8—5081no—
162Q8—5082no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—1271no—
170C2×C4—1271no—
171C2×C4—1271no—
172C2×C4—1271no—
173C2×C4—5081no—
174C2×C4—5081no—
175C2×C4—2542no—
176C2×C4—2542no—
177C2×C4—2541no—
178C2×C4—2541no—
179C2×C4—2541no—
180C2×C4—2541no—
181C2×C4C2×C4×D12711yescharacteristic
182C2×C4D4×D12711yescharacteristic
183C8—41no—
184C8—5081no—
185C22—21no—
186C22—21no—
187C22C22⋊C4×D12711yescharacteristic
188C22#1686123811yescharacteristic
189C22#1686124111yescharacteristic
190C4—21no—
191C4—21no—
192C4—42no—
193C4—41no—
194C4—41no—
195C4—1272no—
196C4—5082no—
197C4—5081no—
198C4—5081no—
199C4—2541no—
200C4—2541no—
201C4—2541no—
202C2—21no—
203C2—21no—
204C2#2254984811yescharacteristic
205C2#2254988311yescharacteristic
206C2#104×D12711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#328×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenter#328D127
hypocenterC127#1506
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
8. #16861208C4semidirect
9. #16861207C4semidirect
13. Q8×Dic127C4semidirect
15. #16861174C4semidirect
104. C127#1506semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225504282
#225499302
#225497212
#225496142
#225496132
#225496122
#328×C1272
#1506127